6v(v+9)=1

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Solution for 6v(v+9)=1 equation:


Simplifying
6v(v + 9) = 1

Reorder the terms:
6v(9 + v) = 1
(9 * 6v + v * 6v) = 1
(54v + 6v2) = 1

Solving
54v + 6v2 = 1

Solving for variable 'v'.

Reorder the terms:
-1 + 54v + 6v2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 54v + 6v2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-0.1666666667 + 9v + v2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 9v + 0.1666666667 + v2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 9v + v2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 9v + v2 = 0 + 0.1666666667
9v + v2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
9v + v2 = 0.1666666667

The v term is 9v.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9v + 20.25 + v2 = 0.1666666667 + 20.25

Reorder the terms:
20.25 + 9v + v2 = 0.1666666667 + 20.25

Combine like terms: 0.1666666667 + 20.25 = 20.4166666667
20.25 + 9v + v2 = 20.4166666667

Factor a perfect square on the left side:
(v + 4.5)(v + 4.5) = 20.4166666667

Calculate the square root of the right side: 4.518480571

Break this problem into two subproblems by setting 
(v + 4.5) equal to 4.518480571 and -4.518480571.

Subproblem 1

v + 4.5 = 4.518480571 Simplifying v + 4.5 = 4.518480571 Reorder the terms: 4.5 + v = 4.518480571 Solving 4.5 + v = 4.518480571 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = 4.518480571 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = 4.518480571 + -4.5 v = 4.518480571 + -4.5 Combine like terms: 4.518480571 + -4.5 = 0.018480571 v = 0.018480571 Simplifying v = 0.018480571

Subproblem 2

v + 4.5 = -4.518480571 Simplifying v + 4.5 = -4.518480571 Reorder the terms: 4.5 + v = -4.518480571 Solving 4.5 + v = -4.518480571 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = -4.518480571 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = -4.518480571 + -4.5 v = -4.518480571 + -4.5 Combine like terms: -4.518480571 + -4.5 = -9.018480571 v = -9.018480571 Simplifying v = -9.018480571

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.018480571, -9.018480571}

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